I need help with this problem:
Consider an experiment that consists of determining the type of
job—either blue-collar or whitecollar—and the political
affiliation—Republican, Democratic, or Independent—of the 15 members
of an adult soccer team. How many outcomes are
in the sample space?
in the event that at least one of the team members is a blue-collar worker?
in the event that none of the team members
considers himself or herself an Independent?
I know that I can star solving this by making a outcome tree or by the counting principle.
So by the counting principle, in the first one I can say that the first experiment is the type of work that has 2 outcomes and the second experiment is the political affiliation that has 3 possible outcomes, thus the number of possible outcomes is calculated by $2\cdot 3=6$, but that is the number of possibilities of just one of them, so I thought that the number of total outcomes should be $6\cdot 15= 90$, but the answer is $6^{15}$, why?
I have the same problem calculating the other two parts.
Best Answer
You are determining the type of work and political affiliation of 15 people. The result of the of the experiment will be a $15$-tuple where the $n$th entry indicates the outcome for the $n$th person, $n=1,2,\dots,15.$ There are $6$ possible value for each entry, as you have shown.
How many possible $15$-tuples are there? There are $6$ values for the first entry. For each of these, there are $6$ values for the second enter, which gives us $36$ possibilities for the first $2$ entries. For each of these, there are $6$ possibilities for the third entry, and so on. In all, there are $6^{15}$ possible outcomes.
If $A,B$ are finite sets, then the cardinality of $A\times B$ is the cardinality of $A$ times the cardinality of $B$, right?